**Introduction**

This is my conceptual note 2 on Ch. 4 in "Ripples in Mathematics"
by A. Jensen & A. la Cour-Harbo. As I noted in my programmatic note 6, when I was trying to programmatically reproduce the multi-resolution CDF(2,2)
transform of a synthetic sinusoid given in Fig. 4.10 in Ch. 4, I had to go back to the CDF(2, 2) recurrences in Ch. 3. While implementing CDF(2, 2), I was confronted with what Jensen & Cour-Harbo call
the

*boundary problem*. I described my tentative solution to the boundary problem in my 6-th programmatic note referenced in the 1st sentence of this post. Since this is work in progress, I will remain flexible and may modify it as I experiment with CDF(2,2) on bigger data. Another conceptual problem I had with CDF(2,2) was implementing the notation used by Jensen and la Cour-Harbo in the recurrences on p. 21 of Ch. 3. In this conceptual note, I discuss my attempt, also tentative, to refine their notation to understand it better and apply my refinement to computing the CDF(2,2) transforms of 2-sample signals with mirror wrapup. The note is based on the notation I introduced in my 1st conceptual note on defining forward DWT recurrences.**Changing CDF(2,2) to CDF(S,2,2,L,J)**

I modified the notation

As shown in the lower part of

*CDF*(2,2) to the notation*CDF*(*S*, 2, 2,*L*,*J*), as shown in**Fig. 1**. The first argument,*S*, denotes the signal, which is a sequence of samples whose length is an integral power of 2. Specifically, the length of*S*is 2^*J*.*L*is the current scale or iteration. It starts at 0, which denotes the original, unmodified signal, and goes up to*J*.Figure 1. CDF(S, 2, 2, L, J) |

**Fig. 1**, when*L*= 0,*CDF*(*S*, 2, 2,*L*,*J*) returns the original signal without any modifications. When*L*> 0,*CDF*(*S*, 2, 2,*L*,*J*) is a modified signal whose first section contains, in LaTeX notation, the elements in*S*^{*L*}_{*J-L*}, then onto*D*^{*L*}_{*J-L*}, and all the way down to*D*^{1}_{*J*-1}.Figure 2. Computing individual elements of CDF(S, 2, 2, L, J) |

**Fig. 2**gives the recurrences for computing the individual elements of

*CDF*(

*S*, 2, 2,

*L*,

*J*) at a given scale

*L*. Let us assume that the mirror wrapup is used and the signal

*S*contains two elements s^{0}_{1}(0) and s^{0}_{1}(1).

*CDF*(

*S*, 2, 2, 1, 1) consists of two values: s^{1}_{0}(0) and d^{1}_{0}(1). To compute d^{1}_{0}(1), we need s^{0}_{1}(0), s^{0}_{1}(1), and s^{0}_{1}(2). The element s^{0}_{1}(2) wraps up to s^{0}^{1}(1).

Figure 3. Computing individual elements of CDF(S, 2, 2, 1, 1) |

**Fig. 3**shows the formulas for the individual elements of

*CDF*(

*S*, 2, 2, 1, 1).

**Examples of Computing CDF(S, 2, 2, 1, 1)**

**Figures 4**,

**5**,

**6**give examples of computing

*CDF*(S, 2, 2, 1, 1) for three 2-sample signals. In

**Fig. 6**, I pilot the notation

*SCDF*(

*S*, 2, 2, 1, 1), which stands for

*standard*CDF without any normalization. I plan to elaborate on it in my next conceptual note where the

*normalized*CDF will be referred to as

*NCDF*.

Figure 4. Computing CDF([1, 4], 2, 2, 1, 1) |

Figure 5. Computing CDF([4, 1], 2, 2, 1, 1) |

Figure 6. Computing CDF([4, 4], 2, 2, 1, 1) |